Checking divisibility of $C^{20}_7-C^{20}_8 +C^{20}_9 -C^{20}_{10}+\dots-C^{20}_{20}$

Question

Let $N=C^{20}_7-C^{20}_8 +C^{20}_9 -C^{20}_{10}+\dots-C^{20}_{20}$. Prove that it is divisible by $3,4,7,19$.

Answer 1

Hint. Since by the binomial theorem $$0=(1-1)^{20}=\sum_{n=0}^{20}\binom{20}{n}(-1)^n=\sum_{n=0}^{6}\binom{20}{n}(-1)^n+\sum_{n=7}^{20}\binom{20}{n}(-1)^n=\sum_{n=0}^{6}\binom{20}{n}(-1)^n-N$$ it follows that \begin{align*} N&=\sum_{n=0}^6\binom{20}{n}(-1)^n\\ &=1-20+190-\frac{20\cdot 19\cdot 18}{3!}+\frac{20\cdot 19\cdot 18\cdot 17}{4!}-\frac{20\cdot 19\cdot 18\cdot 17\cdot 16}{5!}+\frac{20\cdot 19\cdot 18\cdot 17\cdot 16}{6!}\\ &=-19+19\cdot 10-20\cdot 19\cdot 3+5\cdot 19\cdot 3\cdot 17- 19\cdot 3\cdot 17\cdot 16+19\cdot 17\cdot 8\cdot 15. \end{align*} Is it $N$ divisible by $19$ (without evaluating all these product)?

What about $3$, $4$, and $7$?